Collisional model of the drag force of granular impact

Abstract

A dense, dry granular target can cause a free-falling intruding object to come to an abrupt stop as its momentum is lost to the grains. An empirical force law describes this process, characterizing the stopping force as the sum of depth-dependent friction and velocity-dependent inertial drag. However, a complete interpretation of the stopping force, incorporating grain-scale interactions during impact, remains unresolved. Here, the momentum transfer is proposed to occur through sporadic, normal collisions with clusters of high force-carrying grains at the intruder's surface. To test this model in impact experiments, we determine the forces acting on an intruder decelerating through a dense granular medium using high-speed imaging of its trajectory. We vary the geometry of the impacting object to infer intruder-grain interactions. As a result, we connect the inertial drag to the effect of intruder shape based on the proposed collisional model. These impact studies serve as an approach to understand dynamic force transmission in granular media.

Optimal control of plant root tip dynamics in soil

“…To this aim, let us first observe that the soil resistance is relevant just at the tip since the root grows at the apical level by cell elongation and division [26] (figure 1(A)). Also, we considered that penetration tasks, even in artificial systems, are better accomplished by conical or parabolic probe shapes [17,27], similar to root's apexes. And the motion parameters: x(t), the position of the tip; v(t), the velocity; u(t), the acceleration, our control variable; a(t), the axis of symmetry, axis of the tip; α(t), the tip inclination with respect to the vertical axis e 3 .…”

“…Being the soil a complex mixture of solid and fluid structures [29], in addition to the mechanical resistance, our device experiences a dissipative and nonconservative force, called drag force [30,31]. When an object immersed in a fluid has a speed |v| relatively high with respect to the fluid velocity, the drag force is proportional to |v| 2 [17]. On the other hand, at relatively slow speed, the drag force is linear with respect to v [32].…”

“…Furthermore, the experiment proposed in [23], and here used to estimate model parameters, studies a robotic tip that moves with a circumnutating helical pitch comparable to real roots, thus the linearity assumption of the drag force with respect to v is still valid. Therefore, the drag acceleration ād (t) opposing to the tip motion depends on the velocity of the tip v and on the tapering and the lateral surface S l of the device exposed to the interaction with the medium [17]. These properties are expressed in the equation:…”

“…where λ 2 is a parameter estimated in appendix A. The ratio d/(d + h) measures the tapering of the tip that results in a smaller experienced drag force [17]. In addition, for a conical tip of height h and radius r (figure 2(A)), S l = πr √ (h 2 + r 2 ), whereas for a parabolic tip, S l = (πr)((r 2 + 4h 2 ) (3/2) − r 3 )/(6h 2 ).…”

“…Soil is a mixture of organic matter, minerals, gases, liquids and organisms, in which the arrangement of these distinct components defines the soil type, its texture and behaviour in response to external forces [16]. The detailed description of the dynamics of an object moving in soil remains an open problem to date [17]. Secondly, plants have evolved differently in response to the environment, resulting in a wide diversity of genotypes across species and phenotypes over the same species.…”